Question

The motion of a particle connected to a spring is described by x = 10 sin (π*t). At what time (in s) is the potential energy equal to the kinetic energy?a. 0b. 0.25c. 0.50d. 0.79e. 1.0

Answer 1

t = 0.25 seconds ( option b)

Step-by-step explanation:

Assuming that there is no energy loss due to friction , then the total energy E of the system is conserved and is equal to the sum of kinetic K and potential energy V:

E= K + V

where

V= 1/2*k*x² , k= spring constant

K= 1/2*m*v² , v= velocity

but x = 10 sin (π*t)

when the particle reaches its maximum amplitude (x=L=10) , it has no velocity ( can be proven by finding v=dx/dt at t=1/2 for example) , then V=0 and E=K max, then

E=Kmax = 1/2*k*L²

when the kinetic energy is equal to the potential energy K=V , then

E=K+V= K+K=2K = 2*1/2*k*x² = k*x²

since also E=1/2*k*L²

1/2*k*L²=k*x²

x= L/√2 =10 /√2

then

x=10 sin (π*t) = 10 /√2 → t = (1/ π) [sin ⁻¹ (1/√2)] = 0.25

therefore

t = 0.25 seconds